What is the Difference Between Accidental Degeneracy and Normal Degeneracy?

Degeneracy refers to the fact that an energy level is degenerate if it corresponds to two or more different quantum states. There are two forms of degeneracy: accidental degeneracy and normal degeneracy.

Accidental Degeneracy:

• Occurs without any protection by symmetry.
• Results from some special features of the system or the functional form.
• Possibly related to a hidden dynamical symmetry in the system.
• Leads to conserved quantities, which are often not easy to identify.
• Examples include Coulomb and Harmonic Oscillator potentials.

Normal Degeneracy:

• Occurs with the protection by symmetry.
• Results from symmetric transformations of the system.
• The representation obtained from a normal degeneracy is irreducible.
• Examples include the hydrogenic atom with SO(4) symmetry.

In summary, the main difference between accidental and normal degeneracy is that accidental degeneracy occurs without any protection by symmetry, whereas normal degeneracy occurs with the protection by symmetry. Accidental degeneracy is related to the existence of some undiscovered transformations or hidden symmetries in the system, while normal degeneracy is associated with symmetric transformations of the system.

Comparative Table: Accidental Degeneracy vs Normal Degeneracy

Here is a table comparing accidental degeneracy and normal degeneracy:

In summary, accidental degeneracy occurs without any protection by symmetry and is related to special features of the system or the potential, while normal degeneracy occurs with the protection by symmetry and is connected to a hidden dynamical symmetry in the system.